Do you mean human transplants or any type? Also, are you looking for only successful ones? Also, are you talking about internal organs, or would skin transplants count?
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American LukeOct 27 '12 at 21:29

in human ,successful operation and any part of the body,skin or whatever.
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md nthOct 27 '12 at 21:35

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Pre-19th century transplants are not well documented. The first possibly real human transplant was in India. It is unknown if it was successful or not:

The first reasonable account is of the Indian surgeon Sushruta in the 2nd century BC, who used autografted skin transplantation in nose reconstruction rhinoplasty. Success or failure of these procedures is not well documented. -Wikipedia

The first transplants known to be successful occurred in the 16th century:

However, those were transplants from one part of the body to another in the same individual

The first between two individuals was in the 20th century:

In 1905, Eduard Zirm conducted the first successful transplant between two individuals. He transplanted the corneas of an 11-year-old boy into a Czech day laborer. That was the first successful human transplant between two individuals that has been verified.

Note that until the late 1800's, the importance of hand-washing to prevent infections was not widely accepted. Trying to perform a transplant of an internal organ before then would have resulted in so many secondary infections that I can't see it ever having much hope of saving the patient.
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T.E.D.♦Oct 29 '12 at 18:18

True, but that wasn't the point of the question.
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American LukeOct 29 '12 at 18:23

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...which is why I didn't make it an answer. However, I think it backs up the timing you gave here, so I thought it might be helpful.
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T.E.D.♦Oct 29 '12 at 18:25